Library Coq.Numbers.Natural.Abstract.NDiv

Require Import NAxioms NSub NZDiv.

Properties of Euclidean Division

Module Type NDivProp (Import N : NAxiomsSig')(Import NP : NSubProp N).

We benefit from what already exists for NZ

Module Import Private_NZDiv := Nop <+ NZDivProp N N NP.

Ltac auto' := try rewrite <- neq_0_lt_0; auto using le_0_l.

Let's now state again theorems, but without useless hypothesis.

Lemma mod_upper_bound : forall a b, b ~= 0 -> a mod b < b.
Proof. intros. apply mod_bound_pos; auto'. Qed.

Another formulation of the main equation

Lemma mod_eq :
forall a b, b ~=0 -> a mod b == a - b *(a /b ).
Proof.
intros.
symmetry. apply add_sub_eq_l. symmetry.
now apply div_mod.
Qed.

Uniqueness theorems

Theorem div_mod_unique :
forall b q1 q2 r1 r2, r1 r2 
b *q1 +r1 == b *q2 +r2 -> q1 == q2 /\ r1 == r2.
Proof. intros. apply div_mod_unique with b; auto'. Qed.

Theorem div_unique:
forall a b q r, r a == b *q + r -> q == a /b.
Proof. intros; apply div_unique with r; auto'. Qed.

Theorem mod_unique:
forall a b q r, r a == b *q + r -> r == a mod b.
Proof. intros. apply mod_unique with q; auto'. Qed.

Theorem div_unique_exact: forall a b q, b ~=0 -> a == b *q -> q == a /b.
Proof. intros. apply div_unique_exact; auto'. Qed.

A division by itself returns 1

Lemma div_same : forall a, a ~=0 -> a /a == 1.
Proof. intros. apply div_same; auto'. Qed.

Lemma mod_same : forall a, a ~=0 -> a mod a == 0.
Proof. intros. apply mod_same; auto'. Qed.

A division of a small number by a bigger one yields zero.

Theorem div_small: forall a b, a a /b == 0.
Proof. intros. apply div_small; auto'. Qed.

Same situation, in term of modulo:

Theorem mod_small: forall a b, a a mod b == a.
Proof. intros. apply mod_small; auto'. Qed.

Basic values of divisions and modulo.

Lemma div_0_l: forall a, a ~=0 -> 0/a == 0.
Proof. intros. apply div_0_l; auto'. Qed.

Lemma mod_0_l: forall a, a ~=0 -> 0 mod a == 0.
Proof. intros. apply mod_0_l; auto'. Qed.

Lemma div_1_r: forall a, a /1 == a.
Proof. intros. apply div_1_r; auto'. Qed.

Lemma mod_1_r: forall a, a mod 1 == 0.
Proof. intros. apply mod_1_r; auto'. Qed.

Lemma div_1_l: forall a, 1<a -> 1/a == 0.
Proof. exact div_1_l. Qed.

Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.
Proof. exact mod_1_l. Qed.

Lemma div_mul : forall a b, b ~=0 -> (a *b )/b == a.
Proof. intros. apply div_mul; auto'. Qed.

Lemma mod_mul : forall a b, b ~=0 -> (a *b ) mod b == 0.
Proof. intros. apply mod_mul; auto'. Qed.

A modulo cannot grow beyond its starting point.

Theorem mod_le: forall a b, b ~=0 -> a mod b <= a.
Proof. intros. apply mod_le; auto'. Qed.

Lemma div_str_pos : forall a b, 0 0 < a /b.
Proof. exact div_str_pos. Qed.

Lemma div_small_iff : forall a b, b ~=0 -> (a /b ==0 <-> a (a mod b == a <-> a (0<a /b <-> b <=a ).
Proof. intros. apply div_str_pos_iff; auto'. Qed.

As soon as the divisor is strictly greater than 1, the division is strictly decreasing.

Lemma div_lt : forall a b, 0<a -> 1 a /b < a.
Proof. exact div_lt. Qed.

le is compatible with a positive division.

Lemma div_le_mono : forall a b c, c ~=0 -> a <=b -> a /c <= b /c.
Proof. intros. apply div_le_mono; auto'. Qed.

Lemma mul_div_le : forall a b, b ~=0 -> b *(a /b ) <= a.
Proof. intros. apply mul_div_le; auto'. Qed.

Lemma mul_succ_div_gt: forall a b, b ~=0 -> a < b *(S (a /b)).
Proof. intros; apply mul_succ_div_gt; auto'. Qed.

The previous inequality is exact iff the modulo is zero.

Lemma div_exact : forall a b, b ~=0 -> (a == b *(a /b ) <-> a mod b == 0).
Proof. intros. apply div_exact; auto'. Qed.

Some additional inequalities about div.

Theorem div_lt_upper_bound:
 forall a b q, b ~=0 -> a a /b < q.
Proof. intros. apply div_lt_upper_bound; auto'. Qed.

Theorem div_le_upper_bound:
 forall a b q, b ~=0 -> a <= b *q -> a /b <= q.
Proof. intros; apply div_le_upper_bound; auto'. Qed.

Theorem div_le_lower_bound:
 forall a b q, b ~=0 -> b *q <= a -> q <= a /b.
Proof. intros; apply div_le_lower_bound; auto'. Qed.

A division respects opposite monotonicity for the divisor

Lemma div_le_compat_l: forall p q r, 0<q <=r -> p /r <= p /q.
Proof. intros. apply div_le_compat_l. auto'. auto. Qed.

Lemma mod_add : forall a b c, c ~=0 ->
(a + b * c ) mod c == a mod c.
Proof. intros. apply mod_add; auto'. Qed.

Lemma div_add : forall a b c, c ~=0 ->
(a + b * c ) / c == a / c + b.
Proof. intros. apply div_add; auto'. Qed.

Lemma div_add_l: forall a b c, b ~=0 ->
(a * b + c ) / b == a + c / b.
Proof. intros. apply div_add_l; auto'. Qed.

Cancellations.

Lemma div_mul_cancel_r : forall a b c, b ~=0 -> c ~=0 ->
(a *c )/(b *c ) == a /b.
Proof. intros. apply div_mul_cancel_r; auto'. Qed.

Lemma div_mul_cancel_l : forall a b c, b ~=0 -> c ~=0 ->
(c *a )/(c *b ) == a /b.
Proof. intros. apply div_mul_cancel_l; auto'. Qed.

Lemma mul_mod_distr_r: forall a b c, b ~=0 -> c ~=0 ->
(a *c ) mod (b *c ) == (a mod b ) * c.
Proof. intros. apply mul_mod_distr_r; auto'. Qed.

Lemma mul_mod_distr_l: forall a b c, b ~=0 -> c ~=0 ->
(c *a ) mod (c *b ) == c * (a mod b ).
Proof. intros. apply mul_mod_distr_l; auto'. Qed.

Operations modulo.

Theorem mod_mod: forall a n, n ~=0 ->
(a mod n ) mod n == a mod n.
Proof. intros. apply mod_mod; auto'. Qed.

Lemma mul_mod_idemp_l : forall a b n, n ~=0 ->
((a mod n )*b ) mod n == (a *b ) mod n.
Proof. intros. apply mul_mod_idemp_l; auto'. Qed.

Lemma mul_mod_idemp_r : forall a b n, n ~=0 ->
(a *(b mod n )) mod n == (a *b ) mod n.
Proof. intros. apply mul_mod_idemp_r; auto'. Qed.

Theorem mul_mod: forall a b n, n ~=0 ->
(a * b ) mod n == ((a mod n ) * (b mod n )) mod n.
Proof. intros. apply mul_mod; auto'. Qed.

Lemma add_mod_idemp_l : forall a b n, n ~=0 ->
((a mod n )+b ) mod n == (a +b ) mod n.
Proof. intros. apply add_mod_idemp_l; auto'. Qed.

Lemma add_mod_idemp_r : forall a b n, n ~=0 ->
(a +(b mod n )) mod n == (a +b ) mod n.
Proof. intros. apply add_mod_idemp_r; auto'. Qed.

Theorem add_mod: forall a b n, n ~=0 ->
(a +b ) mod n == (a mod n + b mod n ) mod n.
Proof. intros. apply add_mod; auto'. Qed.

Lemma div_div : forall a b c, b ~=0 -> c ~=0 ->
(a /b )/c == a /(b *c ).
Proof. intros. apply div_div; auto'. Qed.

Lemma mod_mul_r : forall a b c, b ~=0 -> c ~=0 ->
a mod (b *c ) == a mod b + b *((a /b ) mod c ).
Proof. intros. apply mod_mul_r; auto'. Qed.

A last inequality:

Theorem div_mul_le:
forall a b c, b ~=0 -> c *(a /b ) <= (c *a )/b.
Proof. intros. apply div_mul_le; auto'. Qed.

mod is related to divisibility

Lemma mod_divides : forall a b, b ~=0 ->
(a mod b == 0 <-> exists c, a == b *c ).
Proof. intros. apply mod_divides; auto'. Qed.

End NDivProp.